Resolving the Loop

[Echognome]

Rather than add on to Fred's thread, I thought it prudent to start a thread on "the Loop" here.

 

As I think the explanations are clearest relying on the well-established theory of games, I thought it best to start with some definitions and work from there. Much of this has been compiled from Wiki for my convenience.

 

STRATEGIES

 

Game - A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies. We can also add a set of information or beliefs available to the players.

 

Strategy - In game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise; this fully determines the player's behaviour. A player's strategy will determine the action the player will take at any stage of the game, for every possible history of play up to that stage.

 

A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation they could face. A player's strategy set is the set of pure strategies available to that player.

 

A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player, even if their strategy set is finite.

 

Of course, one can regard a pure strategy as a degenerate case of a mixed strategy, in which that particular pure strategy is selected with probability 1 and every other strategy with probability 0.

 

DOMINANT AND DOMINATED STRATEGIES

 

B dominates A: choosing B always gives at least as good an outcome as choosing A. There are 2 possibilities:

B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other player(s) do.

B weakly dominates A: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' actions give B at least the same payoff as A.

 

B is dominated by A: choosing B never gives a better outcome than choosing A, no matter what the other player(s) do. There are 2 possibilities:

B is weakly dominated by A: There is at least one set of opponents' actions for which B gives a worse outcome than A, while all other sets of opponents' actions give A at least the same payoff as B. (Strategy A weakly dominates B ).

B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. (Strategy A strictly dominates B ).

 

RATIONALITY

 

Note that a player is considered rational if he does not choose a strictly dominated strategy. One of the first "loops" you hear about is:

 

Player A is rational.

Player B knows A is rational.

Player A knows that B knows that he is rational.

Player B knows that A knows that B knows... etc ad nauseum.

 

In game theory, the best response is the strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response (or one of the best responses) to the other players' strategies.

 

TIMING

 

Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them effectively simultaneous).

 

A sequential game is a game where one player chooses his action before the others choose theirs. Importantly, the later players must have some information of the first's choice, otherwise the difference in time would have no strategic effect. Extensive form representations are usually used for sequential games, since they explicitly illustrate the sequential aspects of a game. Sequential games are often solved by backward induction.

 

I think that is enough for a common set of definitions at this point.

[Echognome]

This simplest form of a loop (in pure strategies) comes from the "Matching Pennies" game.

 

From Wiki:

 

Matching pennies is the name for a simple example game used in game theory. It is the two strategy equivalent of Rock, Paper, Scissors. Matching pennies, also called the Pesky Little Brother Game or Parity Game, is used primarily to illustrate the concept of mixed strategies and a mixed strategy Nash equilibrium.

 

The game is played between two players, Player A and Player B. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), Player A receives one dollar from Player B (+1 for A, -1 for B). If the pennies do not match (one heads and one tails), Player B receives one dollar from Player A (-1 for A, +1 for B). This is an example of a zero-sum game, where one player's gain is exactly equal to the other player's loss.

 

...<snip>

 

This game has no pure strategy Nash equilibrium since there is no pure strategy (heads or tails) that is a best response to a best response. Alternatively, there is no strategy set where neither player would want to switch when told what the other would do. Instead, the unique Nash equilibrium of this game is in mixed strategies: each player chooses heads or tails with equal probability. In this way, each player makes the other indifferent between choosing heads or tails, so neither player has an incentive to try another strategy.

 

Note that this is a simultaneous game and has an equilibrium in mixed strategies, but not pure strategies.

[Echognome]

So, my solution to the Loop is as follows:

 

1) If the opponents ask me what defense I play, then I can tell them my entire strategy set. If they play X, I play Y, for all X (that is reasonable). The opponents can then tell me what they play. The opponents are defined to be limited to playing only strategies allowed by convention regulations. They are allowed to psych (deviate from their agreements), but restricted to the general rules and disclosure relating to psychs.

 

Note that we do not have the Loop of:

 

"We play conventional doubles"

"Then we play a weak NT"

"Then we play penalty doubles"

"Then we play a strong NT"

etc.

 

Because the conversation goes:

 

"We play penalty doubles against weak NT (and 3rd NV NT) and conventional doubles against strong NT."

"Ok. Then we play strong NT."

 

The conversation doesn't go:

 

"Then we play a strong NT, but we're going to psych it often."

 

My feeling is that this is how people would like to play the game.

 

2) My understanding from Ed and David is that the conversation should go differently.

 

"What defense do you play against NT?"

"I won't tell you until you tell me what your agreements are about 1NT first. You have to declare your agreements first before I have to tell you my defense according to regulation."

"We play a strong NT."

"Then we play conventional doubles."

"Then we play ..." <interrupted> "Director!"

[TimG]

Suppose the defenders play penalty doubles when the opponents psych at least 1 in 20* of their 1NT openings, conventional otherwise.

 

The opening side says that 1 in 19 of their 1NT openings is a psych, so penalty doubles are adopted.

 

Opening side knows this so is careful about their psychs and effectively psyches less often than 1 in 20.

 

How do you resolve this?

 

* I picked arbitrary numbers to illustrate, I do not intend to say whether this number of psychs would be legal.

[Echognome]

Suppose the defenders play penalty doubles when the opponents psych at least 1 in 20* of their 1NT openings, conventional otherwise.

 

The opening side says that 1 in 19 of their 1NT openings is a psych, so penalty doubles are adopted.

 

Opening side knows this so is careful about their psychs and effectively psyches less often than 1 in 20.

 

How do you resolve this?

 

* I picked arbitrary numbers to illustrate, I do not intend to say whether this number of psychs would be legal.

I would first applaud the attempted discussion at disclosure.

 

The practical solution is to rely on the general thresholds set by the regulating authorities. I know you picked 1 in 20 to demonstrate a point, but I suppose it's more like 1 in 100 that is legal.

 

If the opening side psychs less than that, I don't see what the problem is. We're not protected against psyching in general are we?

[TimG]

If the opening side psyches less than that as a result of the defense, the defenders don't get to play the defense they want to against the method.

 

That is, the opening side effectively, if not intentionally, psyches less often as a result of the known defense.

 

Perhaps it was wrong to use psyches as the example.

 

Suppose I have loose suit quality requirements for my weak two-bids. The opponents decide to play penalty doubles. I tighten up my suit quality requirements a bit.

 

It sounds like you would resolve this with the defenders stating "we play penalty doubles against loose suit quality requirements; takeout doubles against tight suit quality requirements". So, the opening side picks tighter suit quality requirements. Then the opening side comes across a bad suit that looks good for a preempt and opens a weak two-bid.

 

I guess my question is: how can you be sure of methods when judgment is involved?

 

I do not mean to suggest the the current loop resolution does any better with the problem.

[Echognome]

Tim,

 

I appreciate your input on the matter. My view is that people are unnecessarily entangling the Loop issue (which to me is not an issue) and less than full disclosure. I believe the laws are pretty clear on disclosure. If you feel that they are uneforceable in many cases, then I agree with you. I would go further to say that you cannot stop all forms of cheating either and that's how I view people trying to skate on disclosure issues.

 

So to answer your post directly, I'd say you have the solution. When the opening side then opens a bad weak two-bid and opens it anyway (that isn't a problem in itself), but then his partner plays him for it (let's say he "fields it"), then we do have a problem. But this problem is no different than any other problem involving psyching.

 

I hope that clarifies it.

 

-Gnome

[helene_t]

Psyches that are so frequent that opps need to device defense against them are no longer psyches but partnership agreements. So the problem is with mixed strategies, not with psyches.

 

Purely theoretically, a solution could be to force the 1NT-openers to be explicit about their funny-notrump tendencies, and adhere to them. For example: with a 33(34) 5-9 points I open 1NT if the rank sum of my cards is a multiplum of 10. Then the probability is 10% and opps can chose an adequate defense based on that. If I have a suitable hand and fail to open 1NT (for example because opps have announced to play penalty doubles), then opps can file a recorder form.

 

Or ban mixed strategies for regular partnerships. But that is impractical. When I open a slightly off-shape weak 1NT is first seat I do so on the basis of many factors. I cannot give you the exact algorithm I follow. There will always be borderline hands which I may decide on the basis of how much wine I have had, or whether the facial expression of my LHO suggests he might get a bidding problem if I open 1NT. This must be allowed because otherwise it would only be Ken Rexford and a few of his peers that could abide by the law.

 

The practical solution is this, I think: Write in the laws that mixed strategies are allowed but the key to which alternative to use can't be which defense opponents play. Difficult to enforce because the key is often something like "gut feelings". But at least it would be clear that you can't intentionally invoke the Loop.

[MFA]

So the defense is, in principle, required to produce and disclose a NT defense against every single possible definition of 1NT (including "psyching" frequencies) in every seat and every position? So that the 1NT opener can choose freely?

 

Backwards for sure. And not even possible.

 

And the loop could still be there, if we go deeper ("our defense depends on your defense to our defense - do you play lebensohl etc.?", and so on).

 

The only sensible thing IMO is to insist that bids are declared in 'chronological' order.

[bluejak]

Psyches that are so frequent that opps need to device defense against them are no longer psyches but partnership agreements.

While this is often true, it is certainly not universally true.

 

In England, while psyches are fairly rare, they are more common in these two positions than most others:

 

1X (dbl) 1Y

 

3X p 3Y

 

So I have certain agreements based on the fact that while a specific pair may just about never psyche, in time someone will. As a result I play

 

(1X) dbl (1Y) dbl = 4 cards in Y, 4+ HCP

(1X) dbl (1Y) 2Y = 5+ cards in Y, 4-9 HCP

After

(1X) dbl (1Y)

all bids by us in Y are natural, cue-bids are made in the X suit

 

As for

(3X) p (3Y)

after that I play double as penalties showing Y length: again, later bids in Y are natural.

 

Now this set of agreements - which is fairly common amongst the group with whom I play - is based in part on the possibility of opponents psyching. But it is the general frequency of such psyches, not the frequency by an individual pair, so the assertion quoted does not apply.

[cherdanno]

Short of regulating psychs and mixed strategies, the following might be a more practical solution:

when asked about the defense against an opening bid, the defending side may opt to explain that they use one of several methods, and need not disclose in advance how they choose to pick which method they play. So one could answer "we sometimes play penalty doubles against 1NT, and sometimes it shows a minor or both majors".

[bluejak]

That is certainly illegal.

 

You have a right to know your opponents' methods, and you often need to know them in advance to decide what to do about them. So you may not hide them this way, and why on earth should you?

 

Why not just play bridge? Consider my responses to 1NT over a double: I play Aardvark. But I just about never play against an artificial double. If a pair played an artificial double I might feel I need to know, not so as to decide whether to psyche 1NT or not, but to decide what responses to 1NT mean after an artificial double.

 

What we do not need is complete chaos to stop something happening which only really happens in theory in forums.

[cherdanno]

That is certainly illegal.

I assumed this thread was about how the rules might be changed, I am fully aware that this would not be allowed under current rules.

[WesleyC]

Bluejak: I think you've missed Cherdanno's point.

 

The suggestion is that the defending side fully explains its defensive methods, they just wait until AFTER you have opened 1NT to do so. There's no chaos, the side that opened 1NT knows exactly how to respond (in your case whether or not aardvark applies) and are at no disadvantage. They just can't use the knowledge of the opponents NT defense to change their opening 1NT range.

 

You are correct though - this method is certainly illegal.

[cherdanno]

Bluejak: I think you've missed Cherdanno's point.

 

The suggestion is that the defending side fully explains its defensive methods, they just wait until AFTER you have opened 1NT to do so. There's no chaos, the side that opened 1NT knows exactly how to respond (in your case whether or not aardvark applies) and are at no disadvantage. They just can't use the knowledge of the opponents NT defense to change their opening 1NT range.

 

You are correct though - this method is certainly illegal.

Actually you missed my point too. I was suggesting to say s.th. like "our double is either penalty, or showing a major and a longer minor - we will tell you at the time which we play on any specific hands; feel free to ask about our followup methods if you are interested".

[blackshoe]

The laws say agreements must be disclosed in accordance with the regulations in place. We have such regulations. They work, at least they do when people follow them. Why mess with them?

[WesleyC]

Echognome started this thread with a discussion of Game Theory ideas and then an explanation of a "Matching Pennies" game which bares no resemblance to bridge at all. Consider this game instead:

 

Addition/Subtraction is a non-simultaneous game played between two sides, A and B. First, Player A chooses a positive number, X, less than 100. Then Player B chooses a number Y. The value "X-Y" is then calculated and B must pay A that much. However, if X-Y is negative, B must pay 100 to A.

Clearly A has no advantage in this game. B waits to see X and then chooses the same number himself.

 

Now consider the game if B is forced to play using Echognome's "defense strategy set" idea.

 

Rather than being able to choose a number AFTER Player A, Player B is allowed to choose a (finite) strategy set (based on the value of X) but must reveal it BEFORE A chooses their number.

 

Suppose B goes with something like:

If 0<=X<10 then Y=0

If 10<=X<20 then Y=10

.....

If 90<=X<100 then Y=90

 

Playing against this strategy, A has a positive expectancy. In this case 9.999 units.

 

No matter how large you make B's strategy set, as long as the strategy set remains finite, A will have an advantage in the game.

[awm]

There is a difference between the "bidding game" and the "selecting agreements" game.

 

A mixed strategy in the bidding game is perfectly feasible. A strategy is just a mapping of hands to calls given a particular "auction so far" and randomization could certainly be used and disclosed here. These are the mixed strategies that hrothgar likes to refer to.

 

A mixed strategy in the selecting agreements game would be something like "half the time we play 12-14 notrump and half the time we play 15-17." This makes much less sense, because by their nature the two partners must have the same agreements. If the random bits are shared (i.e. the two players flipped a coin in their hotel room before the session) then they would have to be disclosed. If the random bits are independent then the two partners may be effectively playing different systems and this is really garbage agreements anyway. So a mixed strategy in the selecting agreements game does not really make sense.

 

Of course one can specify conditional strategies. So my defense to notrump could be "Landy if your range is 12-14 and Woolsey if it's 15-17." But if my opponents can also play a conditional strategy, their notrump range could be "15-17 if double is penalty and 12-14 otherwise." This situation will not resolve because both strategies are contingent upon the other side's choice. The game should be either a "one-shot game" (in which both strategies must be specified without being conditional on the opponents' strategy) or a "sequential game" (in which one side must specify strategy first and the other side can then react). The official solution is the latter, requiring the opening side to specify their opening range prior to the opposition specifying their defense.

 

Of course, the issue that sometimes comes up is, in real bridge we are playing multiple games. First we play the specifying agreements game, then we play the bidding game (and then we have to play the cards). Obviously the strategy in the bidding game will depend on the result of the specifying agreements game. The problem is that the requirement of full disclosure restricts the allowed strategy of the bidding game based upon the specifying agreements game -- while occasional deviations from the specified agreements are permitted, frequent high-probability deviations are not. Yet it may be advantageous to deviate frequently based on the strategy selected by the opponents in the specifying agreements game. This is hard to police, but still a violation of law.

[Echognome]

Adam-

 

Very good post. I like how you characterized the different games.

[OleBerg]

The laws say agreements must be disclosed in accordance with the regulations in place. We have such regulations. They work, at least they do when people follow them. Why mess with them?

Some people, me included, think there is a problem (the loop). The proposed changes are meant to solve that problem.

[helene_t]

The laws say agreements must be disclosed in accordance with the regulations in place. We have such regulations. They work, at least they do when people follow them. Why mess with them?

Some people, me included, think there is a problem (the loop). The proposed changes are meant to solve that problem.

Yes, but for most of us the cure is worse than the disease.

 

Most of us never psyche 1NT openings. Nevertheless, we would have to deal with opps playing two different defenses to our 1NT opening and we would have to agree on defenses to both.

[OleBerg]

The laws say agreements must be disclosed in accordance with the regulations in place. We have such regulations. They work, at least they do when people follow them. Why mess with them?

Some people, me included, think there is a problem (the loop). The proposed changes are meant to solve that problem.

Yes, but for most of us the cure is worse than the disease.

 

Most of us never psyche 1NT openings. Nevertheless, we would have to deal with opps playing two different defenses to our 1NT opening and we would have to agree on defenses to both.

Having to make a defense vs a penalty double of 1NT doesn't seem like a tough job to me.

[bluejak]

That is certainly illegal.

I assumed this thread was about how the rules might be changed, I am fully aware that this would not be allowed under current rules.

This forum is for exaplaining the current Laws. I am not arguing how the Laws might be changed, and this forum is unsuitble for doing so.

 

As previously noted, despite the name "The New Laws" I feel that is a suitable forum for discussing ways to change the Laws because we are also loooking forward to the next New Laws.